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NAME _____________________________Math 2 Unit 3
Triangle Congruency In this unit, students will learn to be able to:
- Identify Triangle Congruency- Prove Triangle Congruency- Use Isosceles Triangle Theorem to solve triangles- Solve Equilateral Triangles
Day Lesson Assignment
1 Definition of Congruence
2 Definition of Congruence
3 Triangle Congruence (pictures)
4 Triangle Congruence (pictures)
5 Quiz on Triangle Congruence
6 Triangle Congruence Proofs
7 Triangle Congruence Proofs
8 Quiz on Triangle Congruence Proofs
9 Isosceles Triangle Theorem
10 Isosceles Triangle Theorem
11 Equilateral Triangles
12 Review
13 Test #2
Unit 3 – Packet
1
Congruent Figures: have the same __________________ & same _______________________.Each _____________________ (“matching”) side and angle of congruent figures will also be _______.
2
Example #1:
ABCDE ≅ VWXYZ
Congruent Angles Congruent Sides
Example #2: Given: ABCD≅ EFGH. Complete the followinga) Rewrite the congruence statement in at least 2 more ways.
b) Name all congruent angles
c) Name all congruent sides
We will deal mostly with congruent triangles. Two triangles are congruent if and only if their vertices can be matched up so that the _______________________________ (both angles & sides) are congruent.
Example #3: Given: ∆ HIJ ≅ ∆MNO. Name all congruent sides and all congruent angles.
Write the congruence statement in at least 2 more ways.
Example #4: If ∆ ABC ≅∆ XYZ, and m∠ A=3 x+12, m∠B=3 x+1 and m∠ X=x+44, find x.
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(Points can be named in any consecutive order, but each corresponding vertex must be in the same order for each figure).
What if we wanted to prove that 2 polygons were congruent? What would we need to do?
Because triangles only have three sides, we can take some shortcuts…
I. If all three sides are given, we call this ___________.SSS Postulate: If 3 sides of one triangle are congruent to 3 sides of another triangle, then the triangles are congruent.
II. If 2 sides and the angle BETWEEN those sides are given, we call this _________.SAS Postulate: If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of another triangle, then the triangles are congruent.
III. If 2 angles and the side BETWEEN those angles are given, we call this _________.ASA Postulate: If 2 angles and the included side are congruent to 2 angles and the include side of another triangle, then the triangles are congruent.
IV. If 2 angles and the side NOT BETWEEN those angles are given, we call this _______.AAS Postulate: If 2 angles and their non-included side are congruent to 2 angles and the non-included side of another triangle, then the triangles are congruent.
Example #5: Are the triangles congruent? If so, why?
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Included means _______________
Example #6: IE≅GH ,EF ≅ HF and G is the midpoint of GI .
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Anytime that 2 triangles share a side, think _________________ property!
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Triangle Congruence Picture QuestionsI. If the triangles can be proven congruent, give the reason (SSS, SAS, ASA, or AAS). If there is not enough
information to prove the triangles congruent, write “none.”
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13. 14.
15. 16.
7
Y
P
N
A
B
C D
A
C
B
B
A
K
C
17. 18.
19. 20.
21. 22.
II. Determine whether you can conclude that another triangle is congruent to ∆ABC. If so, complete the congruence statement and give the reason (SSS, SAS, ASA, or AAS). If not, write “none.”
1.
∆ ABC ≅∆¿
by ______________
2.
∆ ABC ≅∆ ¿
by ______________
3.
∆ ABC ≅∆ ¿
by ______________
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BA S
C
6061
60
C B
D A
CB
P
Q
A
CB
PA
Q
B C
AX
Y ZJ
A C
B
4. 5. 6.
∆ ABC ≅∆¿ ∆ ABC ≅∆ ¿ ∆ ABC ≅∆ ¿
by ______________ by ______________ by ______________
7. 8. 9.
∆ ABC ≅∆¿ ∆ ABC ≅∆ ¿ ∆ ABC ≅∆ ¿
by ______________ by ______________ by ______________
For Question #1-4, ΔFIN ≅ ΔWEB
1. Name the three pairs of corresponding sides. _______________
2. Name the three pairs of corresponding angles ______________
3. Is it correct to say F≅ ΔBEW ?_____ Explain why?_________________________________________
4. Is it correct to say F≅ ΔEBW ?_____ Explain why?_________________________________________
For questions #5-9, use the congruent triangles to the right
5. ΔABO ≅ Δ¿ 6. ∡ A≅ ______
7. AO≅ ______ 8. BO=¿ ______
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D
O
C
BA
S
U T
D
V
X K
W
L
JKB A
C
K
L JD F
E
ST
RJ
H
I
T
U
S
M K
L
T
Q S
R
VWK
U
M
What additional information is required in order to know that the triangles are congruent by the given reason?
1. ASA
2. SAS
3. SAS
4. ASA
5. SAS
6. ASA
7. SSS
8. SAS
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Triangle Congruence cont.
Warm-Up
Draw and label a diagram. Then solve for the variable and the missing measure or length.
1. If ∆ BAT ≅ ∆DOG, and m∠B=14 ,m∠G=29 ,∧m∠O=10x+7. Find the value of x m∠O.
x = ___________
m∠O= _________
2. If ∆COW ≅ ∆ PIG, and CO=25 ,CW=18 , IG=23 ,∧PG=7 x−17. Find the value of x and PG.
x = ___________
PG=___________
3. If ∆≝≅ ∆PQR andDE=3 x−10 ,QR=4 x−23 ,∧PQ=2 x+7. Find the value of x and EF.
x = ___________
EF = __________
I. Use the given information and triangle congruence statement to complete the following.11
1. ∆ ABC ≅∆GEO, AB = 4, BC = 6, and AC = 8. What is the length of GO? How do you know?
2. ∆ BAD≅ ∆ LUK ,m∠D=52° ,m∠B=48° ,∧m∠ A=80° .
a. What is the largest angle of ∆ LUK ?
b. What is the smallest angle of ∆ LUK?
3. ∆ SUN ≅ ∆ HOT . ∆ SUN is isosceles. Is there enough information to determine if ∆HOT is isosceles? Explain why or why not.
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A
B
CD
G
E
F
B
A
43
21
D
C
B
A
43
21
D
C
B
A
T
CB
A
C
Y
X
B
A
C
A
B
D
E
II. Complete the congruence statement for each pair of congruent triangles. Then state the reason you are able to determine the triangles are congruent. If you cannot conclude that triangles are congruent, write “none” in the blanks.
1. ∆ EFD≅ ∆¿ 2. ∆ ABC ≅∆ ¿ 3. ∆ LKM ≅ ∆¿
by ________ by ________ by ________
4. ∆ ABC ≅∆ ¿ 5. ∆ ABC ≅∆ ¿
by ________ by ________
III. Use the given information to mark the diagram and any additional congruence you can determine from the diagram. Then complete the triangle congruence statement and give the reason for triangle congruence .
1. 2.
Given: ∠1≅∠3 ,∠2≅∠ 4 Given: ∠ ABD≅∠CBD,∠ ADB≅∠CDB
∆ ABC ≅∆¿ by __________ ∆ ABD≅ ∆¿ by __________
3. 4.
Given: G isthemidpoint of FB∧EA Given: ∠1≅∠3 ,CD ≅ AB∆ ABG≅ ∆¿ by __________ ∆ ABC ≅∆¿ by __________Congruent Triangles Activity
13
H
G
E
F D
J
MK
L
EXAMPLE:1. Name the congruent triangles Δ_ABC_≅Δ_DEF_
2. The triangles are congruent by the _ASA_ postulate
3. List all of the congruent, corresponding sidesAB≅ DEBC ≅ EFAC ≅ DF
4. List all of the congruent, corresponding angles∡ A≅∡D∡B≅∡E
∡C≅∡F5. Draw the congruent triangles
1. Name the congruent triangles Δ_OSU_≅Δ_BCK_
2. The triangles are congruent by the _____ postulate
3. List all of the congruent, corresponding sides
4. List all of the congruent, corresponding angles
5. Draw the congruent triangles
1. Name the congruent triangles Δ_____≅Δ_____
2. The triangles are congruent by the _SAS_ postulate
3. List all of the congruent, corresponding sides
4. List all of the congruent, corresponding angles
5. Draw the congruent triangles
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C
B
A
F
E
D
1. Name the congruent triangles Δ_____≅Δ_____
2. The triangles are congruent by the _____ postulate
3. List all of the congruent, corresponding sidesMR≅ ZILM ≅ MILR≅MZ
4. List all of the congruent, corresponding angles5. Draw the congruent triangles
1. Name the congruent triangles Δ_____≅Δ_____
2. The triangles are congruent by the _____ postulate
3. List all of the congruent, corresponding sides
4. List all of the congruent, corresponding angles∡Q≅∡A∡W ≅∡ S
∡E≅∡D5. Draw the congruent triangles
1. Name the congruent triangles Δ_____≅Δ_____
2. The triangles are congruent by the _____ postulate
3. List all of the congruent, corresponding sides
4. List all of the congruent, corresponding angles
5. Draw the congruent triangles
15
A
L
XI
F
Triangle Congruence Cont. For questions #1-3, use the congruent pentagons below
5. B corresponds to _____
6. BLACK ≅ __________
7. KB = ______ cm
8. If CA⊥LA
, name two right angles
from the figures __________
9. If ΔDEF ≅ ΔRST , m∡D=100 and m∡F=40, name four congruent angles. _______________________
I. ΔPQR≅ Δ ABC. Find the values of x and y.
1. m∠R=5 x+70, m∠C=24 x−25, QR=4 y+2, BC=6 y−8
2. m∠R=90− y, m∠C=13, PR=5 x−10 , AC=32−x
3. PQ=5 x−31, AB=−3 x+1 , QR=−3 y−1, BC=−2 y−2
4. m∠ A=15 y−3, m∠P=12 y+30, PQ=11−x, AB=11 x−1
16
B
L
A
CK
H
O
R
SE
5 cm
4.5 cm
4 cm
5. AB=2 x, PQ=18, BC=11, QR=4 y+3
6. Δ XYZ ≅ ΔMNO, m∠ X=x+10, m∠M=4 x−17, m∠Y=2 y, and m∠N= y+6. Find m∠ X and m∠Y .
II. Solve.
1. The perimeter of ABCD is 85. Find the value of x. Is Δ ABC congruent to Δ ADC? Explain.
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3x + 4 5x - 7
6x - 114x
C
A
B D
18
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Use a Flow Chart to prove each of the following triangles congruent.
1. Given BAD BCD
Prove ABD CBD
2. Given: B E
C is the midpoint of BE & DA
Prove: ABC DEC
3. Given: AC DC
Prove: CAB
CDB
4. Given: AD BC
BA CD
Prove: ABC
CDA
5. Why is this situation not
possible to prove?
6. Given BC EF
CA FD
Prove:
BCA EDF
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Flow Proof Charts Cont.
Class work
Prove: ADB CDB
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Isosceles Triangles
Parts of a Triangle: Triangle – a three-sided polygon
Name –
Sides –
Vertices –
Angles –
Classifying Triangles by Angles:
Acute ∆ Obtuse ∆ Right ∆
Equiangular ∆ -
Classifying Triangles by Sides:
Scalene ∆ Isosceles ∆ Equilateral ∆
Example #1: Find x and the measure of each side of equilateral triangle RST.
23
Example #2: Find x, JM, MN, and JN if ∆JMN is an isosceles triangle
with .
Isosceles Triangle: A triangle with at least __________ sides congruent.
Isosceles Triangle Theorem: If two sides of a triangle are ____________________, then the angles opposite those sides
are ____________________.
Ex:
Example #3: If , , and , what is the measure of ?
24
Theorem: If two angles of a _______________ are congruent, then the sides opposite those angles are
___________________.
Ex:
Example #4:
a.) Name all of the congruent angles.
b.) Name all of the congruent segments.
A triangle is _____________________ if and only if it is ___________________.
Each angle of an equilateral triangle measures ________.
Example #5: ∆EFG is equilateral, and bisects .
a.) Find and .
25
b.) Find x.
26
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Isosceles Triangles cont.Warm-Up1. 2.
3. (x,y)(x+2, -y) 4. Reflect over y= -x.
A’_________ B’_________ C’__________ A’_________ B’_________ C’__________
Find x and y28
70o
x
32o
a
b100o
60o
c d e f
A
B
C
A
C
B
1) 2) 3)
4) 5) Equilateral Triangle 6) Equilateral Triangle
Find x, y and z
7) 8) 9)
29
4
3
4
4
3
9
9
9
Find x (and y) in each figure below.
1. 2.
3.
4. 5.
6.
30
x
x
130o
Unit 3 Study Guide: Triangles
31
For # 1-3, describe each triangle by sides and angles. Use the word bank provided.
Acute ∆ , right ∆ , Obtuse ∆ , scalene ∆ , isosceles ∆ , equilateral ∆1. 2.
a. ________________ a. ________________
b. ________________ b. ________________
3. a.______________________
b. ______________________
For # 4-6, find the given variable(s) for each of the pictures below.
4. t= ___ 5. a = ___
55 72
12t+8 8t-3 4a+12 52
6. y = ___
30
32
7y – 10 y
7. Solve for x, y, and z.
2z - 27o
For # 8 – 13, state whether the triangles are congruent by : SSS, SAS, ASA, AAS, or not possible.
8. 9. 10.
Name the congruent angles and sides for each pair of congruent triangles.
14. Given Δ ABC ≅ ΔDEF ____________ _____________ ____________
____________ _____________ _____________
For #15-19, find the value of x.33
x - 6
x + 10
y + 13
32 x= _______ y = ________ z = _______
z + 15o
46
15. 16.
17. 18.
19.
34
2x - 11 15
3x+12 x + 48
x
4x-12
2x + 30
8045
x7050
x = ______x = ______
x = ______x = ______
x = ______***All tests are cumulative! Don’t forget to review Unit 1 material on Transformations!
35
